Errors in single pixel photography emerging from light collection limits by the bucket detector

In single pixel photography an image is sampled with a programmable optical element like a digital micromirror array or a spatial light modulator that can project an orthogonal base. The light reflected or diffracted is collected by a lens and measured with a photodiode (bucket detector). In this work we demonstrate that single pixel photography that uses sampling bases with non-zero off-diagonal elements (i.e. Hadamard), can be susceptible to errors that emerge from the relative size of the bucket detector area compared with the spatial spread of the Fourier spectrum of the base element that has the the highest spatial frequency. Experiments with a spatial light modulator and simulations using a Hadamard basis show that if the bucket detector area is smaller than between $50-75\%$ of the maximum area spanned by the projected spectrum of the measurement basis, the reconstructed photograph will exhibit cross-talk with the effective phase of the optical system. The phase can be encoded or errors can be introduced in the optical system to demonstrate this effect.